She is smart. At 10, Carlin (Abigail Dylan Harrison) is a tough cookie, loving her practice sessions, determined to be a champion. Coach Jay (Joe Tippett) wants to develop her into a grand-slam winner. She thrives on winning. Her mom, Cyn (Mamie Gummer), thirsts for many things, noticeably a few that conflict with her child’s potential.

Mamie Gummer and Tyee Tilghman in “Our Very Own Carlin McCullough” (Photo by Chris Whitaker)

Thus goes the setup for Amanda Peet’s world premiere play “Our Very Own Carlin McCullough,” onstage at Geffen Playhouse through July 29.

Peet puts much on the stage, yet none of it feels crowded nor artificial. Her plot is well-shaped, with characters that are knowable yet somewhat enigmatic. Each wants something — though over the story, which spans seven years, those wants may change.

Directed by Tyne Rafaeli with finesse to bring out the play’s psychological depths, the evening feels authentic and engagingly purposeful, the actors fleshing but not over-ornamenting their characters, letting the subtext simmer.

Harrison’s little Carlin is adorable, playful yet focused, the type of 10-year-old most of us would want in our families, or perhaps would want to have been.

Her mom drinks so heavily, Carlin calls her “wine breath.” Gummer doesn’t play the drunkenness, though. Instead, she goes deeper, so we see the many facets of what drives a mother to shepherd a talented child.

Cyn says she’s about to lose her job because of all the time off she takes for her child’s matches. But could something else be the issue? She seems to have the warmies for Jay. It’s never clear whether Jay knows, cares, reciprocates. How much of this keeps Jay in the picture, when other coaches might take Carlin further? Or does pushing a child at that age do permanent damage, physical or psychological?

At a match, Cyn meets Stanford University coach Salif (Tyee Tilghman). He puts doubts in Cyn’s mind about Jay’s motives. But Salif has his own motives, which Peet doesn’t shoehorn into her script and which Tilghman doesn’t telegraph to us, leaving just a teasing amount hanging for the audience to ponder.

For much of Act 2, Carlin is 17 years old (Caroline Heffernan). Being a teen, she’s not quite as perfectly adorable, and Heffernan brings that cutting edge of undeserved independence to the stage.

Jay had hoped Carlin would grow tall, as her statuesque mother is. But Carlin remains at a mere 5-foot-3. What other potential has proven unfulfilled, while Carlin develops interests other than playing tennis?

Speaking of which, what’s an appropriate amount of touch between coach and child?

Peet avoids storytelling tawdriness. She trusts the audience to think deeply and intelligently about the action onstage, she lets the characters behave with integrity, given the chance.

Scenic designer Tim Mackabee creates the many nicely detailed settings, on which Rafaeli keeps the action gliding along, helped by having Carlin walk wistfully, thoughtfully, somberly, purposefully across the stage as the set changes.

Lindsay Jones provides original music and sound design, notably that delicious “plop” of the ball on the tennis racket’s sweet spot.